Broadcast, referring to a process of information dissemination in a distributed system whereby a message originating from a certain node is sent to all other nodes in the system, is a very important issue in distributed computing. All-to-all broadcast means the process by which every node broadcasts its certain piece of information to all other nodes. In this paper, we first develop the optimal all-to-all broadcast scheme for the case of one-port communication, which means that each node can only send out one message in one communication step, and then, extend our results to the case of multi-port communication, i.e., k-port communication, meaning that each node can send out k messages in one communication step. We prove that the proposed schemes are optimal for the model considered in the sense that they not only require the minimal number of communication steps, but also incur the minimal number of messages.
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